Definitions
from The American Heritage® Dictionary of the English Language, 5th Edition.
 noun A branch of mathematics in which symbols, usually letters of the alphabet, represent numbers or members of a specified set and are used to represent quantities and to express general relationships that hold for all members of the set.
 noun A set together with a pair of binary operations defined on the set. Usually, the set and the operations simultaneously form both a ring and a module.
from The Century Dictionary.
 noun Formal mathematics; the analysis of equations; the art of reasoning about relations, more especially quantitative relations, by the aid of a compact and highly systematized notation.
 noun Any special system of notation adapted to the study of a special system of relationship: as, “it is an algebra upon an algebra,” Sylvester.
 noun A treatise on algebra.
 noun Its abbreviation is algebra
 noun Algebra of multiple units.
from the GNU version of the Collaborative International Dictionary of English.
 noun (Math.) That branch of mathematics which treats of the relations and properties of quantity by means of letters and other symbols. It is applicable to those relations that are true of every kind of magnitude.
 noun A treatise on this science.
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun uncountable, mathematics A system for computation using letters or other symbols to represent numbers, with rules for manipulating these symbols.
 noun uncountable, mathematics The study of
algebraic structures .  noun countable, mathematics A
universal algebra .  noun countable, algebra An algebraic structure consisting of a
module of acommutative ring along with an additionalbinary operation that isbilinear .  noun countable, set theory, analysis A collection of subsets of a given set, such that this collection contains the empty set, and the collection is closed under
unions andcomplements (and thereby also underintersections anddifferences ).  noun countable, mathematics One of several other types of
mathematical structure .
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 noun the mathematics of generalized arithmetical operations
Etymologies
from The American Heritage® Dictionary of the English Language, 4th Edition
from Wiktionary, Creative Commons Attribution/ShareAlike License
Examples

From the syntactic side, the free Calgebra B on a set X arises as a quotient of the term algebra formed from X (viewed as a set of variables) using the operation symbols and constants common to the algebras of C.

Indeed, the word "algebra" is derived from the title of this book: Kitab alJebr (The Book of Completion) in which he lays out for the first time the rules and steps of solving algebraic equations.

Although your story about not knowing basic algebra is pretty scary ...
Test Scores and Economic Performance, Arnold Kling  EconLog  Library of Economics and Liberty

Like in algebra, we say "Let T be time and D be distance."

Trying to learn these concepts using only simple algebra is actually more difficult in my opinion.
Math and Economics, Arnold Kling  EconLog  Library of Economics and Liberty

Chances are, the algebra 2 teacher has a good idea of what it is really important for students to master in algebra 1.
School Reform, Arnold Kling  EconLog  Library of Economics and Liberty

I'm trying to get you to be actively involved in your own education, to be independent and curious learners in mathematics, even if algebra is never going to be your favorite subject.
Dean Shareski: Personalization vs. Standardization: It's Tough To Do Both

But state officials say the tests  in algebra, English, biology and government  are an important minimum requirement and have improved high school education.
First Click, MarylandObama visits as race accelerates in DC suburbs

Wonderful goals and yet I can help but wonder if algebra is the best vehicle to accomplish these goals.
Dean Shareski: Personalization vs. Standardization: It's Tough To Do Both

While a poor kid is trying to work through an outdated textbook at the kitchen table, his affluent peer across town is being tutored in algebra in her own room.
Comments
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minerva commented on the word algebra
Can also mean a code.
January 9, 2008
minerva commented on the word algebra
I send you enclosed a letter from Mr Lovelace; which, though written in the cursed algebra, I know to be such a one as will show what a queer way he is in; for he read it to us with the air of a tragedian.
Mowbray to Belford, Clarissa by Samuel Richardson
January 9, 2008
artoparts commented on the word algebra
AlJabr means "reunion" from the longer title of the 9th century book by alKhw�?rizmī. The word algorithm derives from his name.
February 12, 2009